Analytical Method For The Traffic Flow Model

Chapter 2 Analytical method for the traffic flow model We discuss the analytic solution of the traffic flow model with a non-linear velocitydensity function by the method of characteristics. We report on the existence and uniqueness of the analytic solution of the traffic flow model. We also present the shock waves of traffic flow.

2.1 Method of characteristics The method of characteristics is a method that can be used to solve the initial value problem for general first order Partial Differential Equations (PDE). We consider the first order linear PDE a ( x, t )

∂u ∂u + b ( x, t ) + c ( x, t ) u = 0 ∂x ∂t

− − − −(2.1)

in two variables along with the initial condition u ( x,0 ) = u 0 ( x ) . The goal of the method of characteristics, when applied to the equation (2.1) is to change co-ordinates from ( x, t ) to a new co-ordinate system ( x 0 , s ) in which the PDE becomes an ordinary differential equation (ODE) along certain curves in the x − t plane. Such curves,

{[ x( s ) , t ( s ) ] : 0 < s < ∞}

along which the solution of the PDE

reduces to an ODE, are called characteristic curves or just the characteristics. The new variable

s

will vary and the new variable x 0 will be constant along the characteristics.

The variable x 0 will change along the initial curve in the

x − t plane. Notice that if we

choose dx = a ( x, t ) − − − −(2.2) ds and

dt = b( x, t ) − − − −( 2.3) ds

then we have du ∂u dx ∂u dt ∂u ∂u = + = a ( x, t ) + b( x, t ) ds ∂x ds ∂t ds ∂x ∂t

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and along the characteristics, the PDE becomes an ODE du + c( x, t )u = 0 − − − −(2.4) ds

The equations (2.2) and (2.3) will be referred to as the characteristic equations.

2.1.1 General strategy The general strategy for applying the method of characteristics to a PDE of the form (2.1) is presented below.

•

Step- 1: First we solve the two characteristic equations (2.2) and (2.3). We find the constant of integration by setting x( 0 ) = x 0 . We now have the transformation from ( x, t ) to ( x 0 , s ) and x = x( x 0 , s ) , t = t ( x 0 , s ) .

•

Step- 2: We solve the ODE (2.4) with initial condition u ( 0 ) = u ( x 0 ) , where x 0 are the initial points on the characteristic curves along the t = 0 axis in the

x − t plane. •

Step- 3: We now have a solution u ( x 0 , s ) . We solve for x 0 and

x

and

s

in terms of

t by using step-1 and substitute these values in u ( x 0 , s ) to get the

solution to the original PDE (2.1) as u ( x, t ) .

2.2 Analytic solution of the macroscopic traffic model The non-linear PDE (1.18) mentioned at section 1.6 can be solved if we know the traffic density at a given initial time, i.e., if we have the traffic density at a given initial time t 0 , we can predict the traffic density for all future time t ≥ t 0 , in principle. Then we are required to solve an initial value problem (IVP) of the form

 ∂ρ ∂  ρ2 + ρ.Vmax 1 − 2 ∂t ∂x  ρ max  ρ ( t 0 , x) = ρ 0 ( x)

  = 0  

-------(2.5)

The IVP (2.5) can be solved by the method of characteristics as follows: The PDE in the IVP (2.5) may be written as

∂ρ ∂q ( ρ ) + =0 ∂t ∂x  ρ2 where q ( ρ ) = ρ.Vmax 1 − 2  ρmax

  

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Now

⇒

∂ρ dq ∂ρ + =0 ∂t dρ ∂x

⇒

 ∂ρ 3ρ 2 + V max  1 − 2  ∂t ρ max 

dρ ∂ρ ∂ρ dx = + =0 dt ∂t ∂x dt

Where

 dx 3ρ 2 = Vmax 1 − 2 dt ρ max  

Equations (2.6) and (2.7) give x( t ) = V max 1 −



 ∂ρ   ∂x = 0 

− − − −(2.6)

   

− − − −(2.7)

3ρ 2 2 ρ max

 t + x 0  

− − − −(2.8)

, which is the characteristics of the IVP (2.5).

t x(t)

0

x

Fig 2.1: The Characteristics for traffic flow model

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Now from equation (2.6) we have dρ =0 dt ∴ρ( x, t ) = c

− − − −(2.9)

(

)

Since the characteristics through ( x, t ) also passes through x 0 ,0 and ρ( x, t ) = c is constant on this curve, so we use the initial condition to write

c = ρ ( x , t ) = ρ ( x 0 ,0 ) = ρ 0 ( x 0 )

− − − −(2.10)

Equation (2.9) and (2.10) yield

ρ ( x, t ) = ρ 0 ( x 0 )

− − − −(2.11)

Using equation (2.8), (2.11) takes the form

  3ρ 2   ρ ( x, t ) = ρ 0  x − Vmax 1 − 2 t     ρ max   

− − − −(2.12)

This is the analytic solution of the IVP (2.5). This solution is in implicit form because ρ also appears in right side. It is much more difficult to transform into explicit form. Therefore, there is a demand of some efficient numerical techniques for solving the IVP (2.5). To check this solution, we note that

V ∂ρ ∂ρ  ′ = ρ 0 1 + 6tρ max  2 ∂x ∂x  ρ max  ∂ρ ⇒ = ∂x

and

′ ρ0 ′ 6tρVmax 1− ρ0 2 ρ max

 ∂ρ 3ρ 2 ′  = ρ 0 − Vmax 1 − 2 ∂t   ρ max

− − − − (2.13)

 6tρVmax ∂ρ  +  2  ∂t  ρ max 

 3ρ 2  ′ − ρ 0 V max 1 − 2  ρ max  ∂ρ  ⇒ = ∂t ′ 6tρV max 1 − ρ0 2 ρ max

− − − −(2.14)

  3ρ 2    The equations (2.13) and (2.14) imply that ρ ( x, t ) = ρ 0 x − V max  1 − 2 t     ρ max   

satisfies

the quasilinear form of the IVP (2.5).

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2.3 Development of singularities for analytic solution Since the solution (2.12) generally defines ρ implicitly, we need to find circumstances under which we can solve this equation for ρ . We have used (2.13) & (2.14) to solve for the partial derivatives for ρ . These equations allow us to make several observations.

 Suppose that

6 ρV max

ρ

∂ρ ∂ρ > 0 for all ρ . in this case, and are bounded

2 max

∂t

for all t if and only if ρ0 ′ ≥ 0 for all

 Next, we suppose that

6 ρV max

ρ

2 max

x.

∂ρ ∂ρ < 0 for all ρ . in this case, and are ∂t

bounded for all t if and only if ρ0 ′ ≤ 0 for all

 Next, we suppose that

and

6 ρV max

ρ

2 max

∂x

∂x

x.

∂ρ ′ = 0 for all ρ . in this case, = −ρ0 V max

∂ρ ′ = ρ0 are bounded for all ∂x

∂t

t.

2.4 Existence & Uniqueness of the solution The Cauchy-Kovaleveskaya theorem is the main local existence and uniqueness theorem for analytic partial differential equation (PDE) associated with Cauchy initial value problem. This idea was given by Augustin Cauchy (1842) and Sophie Kovalevskaya (1875). Since the IVP (2.5) is a Cauchy IVP, so to discuss the existence and uniqueness of the IVP (2.5), we can readily apply the Cauchy-Koveleveskaya theorem.

2.4.1 The Cauchy-Kovaleveskaya Theorem For a system of

k

partial differential equations in

k

unknown functions

u1 ( x, x 0 ),..., u k ( x, x 0 ) , of the form

 ∂ mui ∂ m1 + ...+ mn u  = F x , x , u , i 0 m m ∂x 0m ∂x 0 0 ...∂x n n  where

i = 1,..., k ,

x = ( x1 ,...., x n ) ,

 ,  

− − − − (2.15)

u = ( u1 ,..., u k ) ,

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n

∑m j =0

j

≤ m,

m 0 < m,

m ≥ 1,

The Cauchy-Kovalevskaya theorem states the following: The Cauchy problem posed by the initial data

∂ j ui ∂x 0j where

= φ ij ( x ) ,

i = 1,..., k ,

j = 0,..., m − 1

− − − −(2.16)

σ

σ = { ( x, x 0 ) : x 0 = 0, ∀x ∈ Ω 0 } is the initial surface of the data φij , has a unique

analytic solution u ( x, x 0 ) in some domain Ω in ( x 0 , x ) -space containing Ω 0 × { x 0 } , if Fi and φij are analytic functions of all their arguments. The proof of the theorem does not present here.

2.4.2 Existence & uniqueness of the traffic flow model As mentioned at 2.2 the IVP (2.5) may also be written as

 ∂ρ 3ρ 2 + Vmax 1 − 2 ∂t ρ max 

ρ ( t 0 , x) = ρ 0 ( x)  3ρ 2  V 1 − In that case, max  2 ρ max 

 ∂ρ   ∂x = 0 

---------(2.17)

  is analytic. So the Cauchy-Kovaleveskaya theorem can  

be applied for the IVP (2.17) (Cauchy problem). The Cauchy-kovaleveskaya theorem guarantees that the Cauchy problem (2.17) has a unique analytic solution which is based on [27].

2.5 Shock waves of traffic flow When fast traffic catches up with slow traffic, a shock wave will form and travel at the speed of. Vehicles reduce speed immediately after crossing the shock. The characteristic speed for the equation in the IVP (2.17) is

 3ρ 2 q′( ρ ) = Vmax 1 − 2 ρmax 

  

while the shock speed for a jump from ρl to ρr is

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S=

q( ρ l ) − q ( ρ r ) ρl − ρr

  1 = V max 1 − 2 ρ l2 + ρ r2 + ρ l ρ r   ρ max 

(

)

− − − −(2.18)

We consider the initial data

 ρ l ;x< 0 ρ ( x ,0 ) =   ρ r ;x > 0 where 0 < ρ l < ρ r < ρ max . Then the solution is a shock wave traveling with speed S given by (2.18). Note that although V ( ρ) ≥ 0 , the shock speeds can be either positive or negative depending on

ρl and ρr . We consider the case ρr = ρmax

and ρl < ρmax . Then S < 0 and the shock

propagates to the left. This models the situation in which cars moving at speed Vl > 0 unexpectedly encounter a bumper to bumper traffic jam and slam on their brakes, instantaneously reducing their velocity to 0 while the density jumps from ρl to ρmax . This discontinuity occurs at the shock, and clearly the shock location moves to the left as more cars join the traffic jam. This is illustrated in fig-2.2, where the vehicle trajectories are sketched.

t↑

ρl

0

x

ρr 1 2

Fig 2.2: Traffic jams Shock wave, with data ρl = ρ max ; ρ r = ρ max

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Note that by using Godunova’s scheme, we can solve the shock wave numerically. This is not present here.

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